Quantum Mechanics

Introduction

This course is given to all second year physicists
and is examined on paper A3 at the end of the second year.
For the academic year 2014/2015, the lecture course begins in Michaelmas Term 2014 (12 lectures) with the remainder of the lectures in Hilary Term 2015 (15 lectures).

In this course we introduce the subject of
quantum mechanics.

Synopsis

Probabilities and probability amplitudes. Interference, state
vectors and the bra-ket notation, wavefunctions. Hermitian operators
and physical observables, eigenvalues and expectation values. The
effect of measurement on a state; collapse of the wave
function. Successive measurements and the uncertainty relations. The
relation between simultaneous observables, commutators and complete
sets of states.

The time-dependent Schrödinger equation. Energy eigenstates and the time-independent Schroedinger equation. The time evolution of a system not in an energy eigenstate. Wave packets in position and momentum space.

Probability current density.

Wave function of a free particle and its relation to de Broglie's hypothesis and Planck's relation. Particle in one-dimensional square-well potentials of finite and infinite depth. Scattering off, and tunnelling through, a one-dimensional square potential barrier. Circumstances in which a change in potential can be idealised as steep; [Non examinable: Use of the WKB approximation.]

The simple harmonic oscillator in one dimension by operator methods. Derivation of energy eigenvalues and eigenfunctions and explicit forms of the eigenfunctions for n=0,1 states.

Amplitudes and wave functions for a system of two particles. Simple examples of entanglement.

Commutation rules for angular momentum operators including raising
and lowering operators, their eigenvalues (general derivation of the
eigenvalues of L2 and L not required), and
explicit form of the spherical harmonics for l=0,1
states. Rotational spectra of simple diatomic molecules.

Representation of spin-1/2 operators by Pauli matrices. The magnetic moment of the electron and precession in a homogeneous magnetic field. The Stern-Gerlach experiment. The combination of two spin-1/2 states into S=0,1; [non-examinable: Derivation of states of well-defined total angular momentum using raising and lowering operators]. Rules for combining angular momenta in general (derivation not required). [Non-examinable: Spectroscopic notation.]

Hamiltonian for the gross structure of the hydrogen atom. Centre
of mass motion and reduced particle. Separation of the kinetic-energy
operator into radial and angular parts. Derivation of the allowed
energies; principal and orbital angular-momentum quantum numbers;
degeneracy of energy levels. Functional forms and physical
interpretation of the wavefunctions for n<3.